How Isaac Newton discovered the binomial power series (2022)(quantamagazine.org)
quantamagazine.org
How Isaac Newton discovered the binomial power series (2022)
https://www.quantamagazine.org/how-isaac-newton-discovered-the-binomial-power-series-20220831/
18 comments
Thanks! This sounds interesting and I looked it up.
To save everyone else a few seconds: https://www.jstor.org/stable/pdf/2318338.pdf
To save everyone else a few seconds: https://www.jstor.org/stable/pdf/2318338.pdf
Abhyankar was a master of "unlearning". He defended the importance and relevance of high-school mathematics for "deep results" all his life.
I think the significance of the Epistola Posterior only really emerged decades after it was written, when the bitter Newton–Leibniz priority controversy erupted. It served as Newton’s key evidence, by revealing that he had the main ideas well before Leibniz’s first publications (1684). The letter bolstered the Royal Society’s eventual judgment favoring Newton’s independent discovery.
I honestly feel that with entertainment as accessible as it is today, almost any mind that could come up with this today would be immediately distracted away.
I don’t. My 11-year-old son managed, without knowing algebra, to find the square root of i without guidance. His technique was not dissimilar to what Newton described. And while he’s bright, I don’t know that I would put him in a class with Newton. Even with distractions available, people are still able to focus on questions like these and come up with solutions.
This stuff always makes me chuckle. Magnus Carlsen learned chess without guidance, modulo his father being a rated player.
> My 11-year-old son managed, without knowing algebra, to find the square root of i without guidance.
I'm curious what this means. To my mind, "without guidance" and also without knowing algebra, he wouldn't know what i was, and would therefore be unable to do any computation with it.
I'm curious what this means. To my mind, "without guidance" and also without knowing algebra, he wouldn't know what i was, and would therefore be unable to do any computation with it.
He knew that i was the square root of 1 and he had learned some really basic skills like the distributive law, so he managed to work out that (a+bi)(a+bi) = a²-b²+2abi, but it was entirely self-directed on his part. I had taught him some how to solve simple linear equations but he didn’t have the skills to be able to work out that he needed a=b and 2ab=1 so there were some trial and error attempts at finding values that worked.
(I did the same thing as he did, but in high school sophomore year while bored in health class and with a bit more sophistication mathematically than what he had.)
(I did the same thing as he did, but in high school sophomore year while bored in health class and with a bit more sophistication mathematically than what he had.)
lots of "math processing error". perhaps just render the formulas?
I dont have such errors in neither firefox or safari, all latex is rendered fine; but some of the equation blocks are images vs processed latex, and this is annoying with trying to render a dark theme with dark reader.
I'm not getting any processing errors. Seems to render fine on both Chromium and Firefox.
If you have a js blocker, try blocking the js from disqus.com.
yep, on Google chrome it all appears as unrendered math latex variables. This article is unreadable because of this.
Works fine on my chrome?
Works for me in Firefox; fails in Safari.
I get those "math processing errors" in Firefox, after some time. Some of the error messages, which appear right before those:
GET https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0 NS_ERROR_CORRUPTED_CONTENT
The resource from “https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0” was blocked due to MIME type (“text/html”) mismatch (X-Content-Type-Options: nosniff).
Loading failed for the <script> with source “https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0”. how-isaac-newton-discovered-the-binomial-power-series-20220831:1:1
Uncaught TypeError: c.FONTDATA.FONTS.MathJax_Main[8212][5] is undefined
Works fine for me in Safari.
> The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester.
and goes on to write:
> Personal Experience 1. In my Harvard dissertation (1956, [2]) I proved resolution of singularities of algebraic surfaces in nonzero characteristic. There I used a mixture of high-school and college algebra. After ten years, I understood the Binomial Theorem a little better and thereby learned how to replace some of the college algebra by high-school algebra; that enabled me to prove resolution for arithmetical surfaces (1965, [4]). Then replacing some more college algebra by high-school algebra enabled me to prove resolution for three-dimensional algebraic varieties in nonzero characteristic (1966, [5]). But still some college algebra has remained.
> I am convinced that if one can decipher the mysteries of the Binomial Theorem and learn how to replace the remaining college algebra by high-school algebra, then one should be able to do the general resolution problem. Indeed, I could almost see a ray of light at the end of the tunnel. But this process of unlearning college algebra left me a bit exhausted; so I quit!